n^2-3n-1=0

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Solution for n^2-3n-1=0 equation:


Simplifying
n2 + -3n + -1 = 0

Reorder the terms:
-1 + -3n + n2 = 0

Solving
-1 + -3n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -3n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + -3n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -3n + n2 = 0 + 1
-3n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
-3n + n2 = 1

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 1 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 1 + 2.25

Combine like terms: 1 + 2.25 = 3.25
2.25 + -3n + n2 = 3.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(n + -1.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

n + -1.5 = 1.802775638 Simplifying n + -1.5 = 1.802775638 Reorder the terms: -1.5 + n = 1.802775638 Solving -1.5 + n = 1.802775638 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 1.802775638 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 1.802775638 + 1.5 n = 1.802775638 + 1.5 Combine like terms: 1.802775638 + 1.5 = 3.302775638 n = 3.302775638 Simplifying n = 3.302775638

Subproblem 2

n + -1.5 = -1.802775638 Simplifying n + -1.5 = -1.802775638 Reorder the terms: -1.5 + n = -1.802775638 Solving -1.5 + n = -1.802775638 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -1.802775638 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -1.802775638 + 1.5 n = -1.802775638 + 1.5 Combine like terms: -1.802775638 + 1.5 = -0.302775638 n = -0.302775638 Simplifying n = -0.302775638

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.302775638, -0.302775638}

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